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<li class="toctree-l3"><a class="reference internal" href="#ot.sliced.expected_sliced_plan"><code class="docutils literal notranslate"><span class="pre">expected_sliced_plan()</span></code></a></li>
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<section id="module-ot.sliced">
<span id="ot-sliced"></span><h1>ot.sliced<a class="headerlink" href="#module-ot.sliced" title="Link to this heading"></a></h1>
<p>Solvers related to (balanced) sliced transport.</p>
<dl class="py function">
<dt class="sig sig-object py" id="ot.sliced.expected_sliced_plan">
<span class="sig-prename descclassname"><span class="pre">ot.sliced.</span></span><span class="sig-name descname"><span class="pre">expected_sliced_plan</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X_s</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">X_t</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">metric</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'sqeuclidean'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n_projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">beta</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dense</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">batch_size</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">log</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/sliced/_sliced_plans.html#expected_sliced_plan"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.sliced.expected_sliced_plan" title="Link to this definition"></a></dt>
<dd><p>Computes the Expected Sliced cost and plan between two datasets <cite>X_s</cite> and
<cite>X_t</cite> of shapes <cite>(ns, d)</cite> and <cite>(nt, d)</cite>. Given a set of <cite>n_projections</cite> projection
directions, the expected sliced plan is obtained by averaging the <cite>n_projections</cite>
1d optimal transport plans between the projections of <cite>X_s</cite> and <cite>X_t</cite> on each
direction. Expected Sliced was introduced in [87] and further studied in
[86].</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The computation ignores potential ambiguities in the projections: if
two points from a same measure have the same projection on a direction,
then multiple sorting permutations are possible. To avoid combinatorial
explosion, only one permutation is retained: this strays from theory in
pathological cases.</p>
</div>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>Tensorflow and jax only returns dense plans, as they do not support well sparse matrices.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>X_s</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>ns</em><em>, </em><em>d</em><em>)</em>) – The first set of vectors.</p></li>
<li><p><strong>X_t</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>nt</em><em>, </em><em>d</em><em>)</em>) – The second set of vectors.</p></li>
<li><p><strong>a</strong> (<em>ndarray</em><em> of </em><em>float64</em><em>, </em><em>shape</em><em> (</em><em>ns</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – Source histogram (default is uniform weight)</p></li>
<li><p><strong>b</strong> (<em>ndarray</em><em> of </em><em>float64</em><em>, </em><em>shape</em><em> (</em><em>nt</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – Target histogram (default is uniform weight)</p></li>
<li><p><strong>projections</strong> (<em>shape</em><em> (</em><em>dim</em><em>, </em><em>n_projections</em><em>)</em><em>, </em><em>optional</em>) – Projection matrix (n_projections and seed are not used in this case).
Default is None</p></li>
<li><p><strong>metric</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)"><em>str</em></a><em>, </em><em>optional</em><em> (</em><em>default='sqeuclidean'</em><em>)</em>) – Metric to be used. Only works with either of the strings
<cite>‘sqeuclidean’</cite>, <cite>‘minkowski’</cite>, <cite>‘cityblock’</cite>, or <cite>‘euclidean’</cite>.</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em><em> (</em><em>default=2</em><em>)</em>) – The p-norm to apply for if metric=’minkowski’</p></li>
<li><p><strong>n_projections</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – The number of projection directions. Required if projections is None.</p></li>
<li><p><strong>seed</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – The seed for the random number generator for sampling projections, in case
projections is None. Default is None.</p></li>
<li><p><strong>beta</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em>) – Inverse-temperature parameter which weights each projection’s
contribution to the expected plan. Default is 0 (uniform weighting).</p></li>
<li><p><strong>dense</strong> (<em>boolean</em><em>, </em><em>optional</em><em> (</em><em>default=True</em><em>)</em>) – If True, returns <span class="math notranslate nohighlight">\(\gamma\)</span> as a dense ndarray of shape (ns, nt).
Otherwise returns a sparse representation using scipy’s <cite>coo_matrix</cite>
format.</p></li>
<li><p><strong>batch_size</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – If specified, compute the distance in batches of size <cite>batch_size</cite> to
avoid memory issues for large datasets. Default is None (no batching).</p></li>
<li><p><strong>log</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.14)"><em>bool</em></a><em>, </em><em>optional</em>) – If True, returns additional logging information. Default is False.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>plan</strong> (<em>ndarray, shape (ns, nt) or coo_matrix if dense is False</em>) – Optimal transportation matrix for the given parameters.</p></li>
<li><p><strong>cost</strong> (<em>float</em>) – The cost associated to the optimal permutation.</p></li>
<li><p><strong>log_dict</strong> (<em>dict, optional</em>) – A dictionary containing intermediate computations for logging purposes.
Returned only if <cite>log</cite> is True.</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id1" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>86<span class="fn-bracket">]</span></span>
<p>Tanguy, E., Chapel, L., Delon, J. (2025). Sliced Transport
Plans. arXiv preprint 2506.03661.</p>
</aside>
<aside class="footnote brackets" id="id2" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>87<span class="fn-bracket">]</span></span>
<p>Liu, X., Diaz Martin, R., Bai Y., Shahbazi A., Thorpe M., Aldroubi
A., Kolouri, S. (2024). Expected Sliced Transport Plans.
International Conference on Learning Representations.</p>
</aside>
</aside>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">ot</span>
<span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">3.</span><span class="p">,</span><span class="mf">3.</span><span class="p">],</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span><span class="mf">1.</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">2.</span><span class="p">,</span><span class="mf">2.5</span><span class="p">],</span> <span class="p">[</span><span class="mf">3.</span><span class="p">,</span><span class="mf">2.</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">projections</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">plan</span><span class="p">,</span> <span class="n">cost</span> <span class="o">=</span> <span class="n">ot</span><span class="o">.</span><span class="n">expected_sliced_plan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">projections</span><span class="o">=</span><span class="n">projections</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">plan</span>
<span class="go">array([[0.25, 0.25],</span>
<span class="go"> [0.25, 0.25]])</span>
<span class="gp">>>> </span><span class="n">cost</span>
<span class="go">2.625</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.sliced.get_projections_sphere">
<span class="sig-prename descclassname"><span class="pre">ot.sliced.</span></span><span class="sig-name descname"><span class="pre">get_projections_sphere</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">d</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n_projections</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">backend</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">type_as</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/sliced/_utils.html#get_projections_sphere"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.sliced.get_projections_sphere" title="Link to this definition"></a></dt>
<dd><p>Generates n_projections samples from the uniform distribution on the Stiefel manifold of dimension <span class="math notranslate nohighlight">\(d\times 2\)</span>: <span class="math notranslate nohighlight">\(\mathbb{V}_{d,2}=\{X \in \mathbb{R}^{d\times 2}, X^TX=I_2\}\)</span></p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>d</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a>) – dimension of the space</p></li>
<li><p><strong>n_projections</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a>) – number of samples requested</p></li>
<li><p><strong>seed</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em> or </em><em>RandomState</em><em>, </em><em>optional</em>) – Seed used for numpy random number generator</p></li>
<li><p><strong>backend</strong> – Backend to use for random generation</p></li>
<li><p><strong>type_as</strong> (<em>optional</em>) – Type to use for random generation</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong></p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray, shape (n_projections, d, 2)</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">n_projections</span> <span class="o">=</span> <span class="mi">100</span>
<span class="gp">>>> </span><span class="n">d</span> <span class="o">=</span> <span class="mi">5</span>
<span class="gp">>>> </span><span class="n">projs</span> <span class="o">=</span> <span class="n">get_projections_sphere</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">n_projections</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s2">"nij, nik -> njk"</span><span class="p">,</span> <span class="n">projs</span><span class="p">,</span> <span class="n">projs</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.sliced.get_random_projections">
<span class="sig-prename descclassname"><span class="pre">ot.sliced.</span></span><span class="sig-name descname"><span class="pre">get_random_projections</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">d</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n_projections</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">backend</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">type_as</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/sliced/_utils.html#get_random_projections"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.sliced.get_random_projections" title="Link to this definition"></a></dt>
<dd><p>Generates n_projections samples from the uniform on the unit sphere of dimension <span class="math notranslate nohighlight">\(d-1\)</span>: <span class="math notranslate nohighlight">\(\mathcal{U}(\mathcal{S}^{d-1})\)</span></p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>d</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a>) – dimension of the space</p></li>
<li><p><strong>n_projections</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a>) – number of samples requested</p></li>
<li><p><strong>seed</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em> or </em><em>RandomState</em><em>, </em><em>optional</em>) – Seed used for numpy random number generator</p></li>
<li><p><strong>backend</strong> – Backend to use for random generation</p></li>
<li><p><strong>type_as</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#type" title="(in Python v3.14)"><em>type</em></a><em>, </em><em>optional</em>) – Type of the returned array</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>out</strong> – The uniform unit vectors on the sphere</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray, shape (d, n_projections)</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">n_projections</span> <span class="o">=</span> <span class="mi">100</span>
<span class="gp">>>> </span><span class="n">d</span> <span class="o">=</span> <span class="mi">5</span>
<span class="gp">>>> </span><span class="n">projs</span> <span class="o">=</span> <span class="n">get_random_projections</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">n_projections</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">square</span><span class="p">(</span><span class="n">projs</span><span class="p">),</span> <span class="mi">0</span><span class="p">),</span> <span class="mf">1.</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.sliced.linear_sliced_wasserstein_sphere">
<span class="sig-prename descclassname"><span class="pre">ot.sliced.</span></span><span class="sig-name descname"><span class="pre">linear_sliced_wasserstein_sphere</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X_s</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">X_t</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n_projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">50</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">log</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/sliced/_spherical_sliced.html#linear_sliced_wasserstein_sphere"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.sliced.linear_sliced_wasserstein_sphere" title="Link to this definition"></a></dt>
<dd><p>Computes the linear spherical sliced wasserstein distance from <a class="reference internal" href="#references-lssot"><span class="std std-ref">[79]</span></a>.</p>
<p>General loss returned:</p>
<div class="math notranslate nohighlight">
\[\mathrm{LSSOT}_2(\mu, \nu) = \left(\int_{\mathbb{V}_{d,2}} \mathrm{LCOT}_2^2(P^U_\#\mu, P^U_\#\nu)\ \mathrm{d}\sigma(U)\right)^{\frac12},\]</div>
<p>where <span class="math notranslate nohighlight">\(\mu,\nu\in\mathcal{P}(S^{d-1})\)</span> are two probability measures on the sphere, <span class="math notranslate nohighlight">\(\mathrm{LCOT}_2\)</span> is the linear circular optimal transport distance,
and <span class="math notranslate nohighlight">\(P^U_\# \mu\)</span> stands for the pushforwards of the projection <span class="math notranslate nohighlight">\(\forall x\in S^{d-1},\ P^U(x) = \frac{U^Tx}{\|U^Tx\|_2}\)</span>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>X_s</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_a</em><em>, </em><em>dim</em><em>)</em>) – Samples in the source domain</p></li>
<li><p><strong>X_t</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_b</em><em>, </em><em>dim</em><em>)</em><em>, </em><em>optional</em>) – Samples in the target domain. If None, computes the distance against
the uniform distribution on the sphere.</p></li>
<li><p><strong>a</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_a</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – samples weights in the source domain</p></li>
<li><p><strong>b</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_b</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – samples weights in the target domain</p></li>
<li><p><strong>n_projections</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – Number of projections used for the Monte-Carlo approximation</p></li>
<li><p><strong>projections</strong> (<em>shape</em><em> (</em><em>n_projections</em><em>, </em><em>dim</em><em>, </em><em>2</em><em>)</em><em>, </em><em>optional</em>) – Projection matrix (n_projections and seed are not used in this case)</p></li>
<li><p><strong>seed</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em> or </em><em>RandomState</em><em> or </em><em>None</em><em>, </em><em>optional</em>) – Seed used for random number generator</p></li>
<li><p><strong>log</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.14)"><em>bool</em></a><em>, </em><em>optional</em>) – if True, linear_sliced_wasserstein_sphere returns the projections used
and their associated LCOT.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>cost</strong> (<em>float</em>) – Linear Spherical Sliced Wasserstein Cost</p></li>
<li><p><strong>log</strong> (<em>dict, optional</em>) – log dictionary return only if log==True in parameters</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">ot</span>
<span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="n">n_samples_a</span> <span class="o">=</span> <span class="mi">20</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="p">(</span><span class="n">n_samples_a</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="n">X</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">X</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">ot</span><span class="o">.</span><span class="n">linear_sliced_wasserstein_sphere</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="go">0.0</span>
</pre></div>
</div>
<p class="rubric" id="references-lssot">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id3" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>79<span class="fn-bracket">]</span></span>
<p>Liu, X., Bai, Y., Martín, R. D., Shi, K., Shahbazi, A., Landman,
B. A., Chang, C., & Kolouri, S. (2025). Linear Spherical Sliced Optimal
Transport: A Fast Metric for Comparing Spherical Data. International
Conference on Learning Representations.</p>
</aside>
</aside>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.sliced.max_sliced_wasserstein_distance">
<span class="sig-prename descclassname"><span class="pre">ot.sliced.</span></span><span class="sig-name descname"><span class="pre">max_sliced_wasserstein_distance</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X_s</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">X_t</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n_projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">50</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">log</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">scaler</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/sliced/_sliced_distances.html#max_sliced_wasserstein_distance"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.sliced.max_sliced_wasserstein_distance" title="Link to this definition"></a></dt>
<dd><p>Computes a Monte-Carlo approximation of the max p-Sliced Wasserstein distance</p>
<div class="math notranslate nohighlight">
\[\mathcal{Max-SWD}_p(\mu, \nu) = \underset{\theta \in
\mathcal{U}(\mathbb{S}^{d-1})}{\max} [\mathcal{W}_p^p(\theta_\#
\mu, \theta_\# \nu)]^{\frac{1}{p}}\]</div>
<p>where :</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(\theta_\# \mu\)</span> stands for the pushforwards of the projection <span class="math notranslate nohighlight">\(\mathbb{R}^d \ni X \mapsto \langle \theta, X \rangle\)</span></p></li>
</ul>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>X_s</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_a</em><em>, </em><em>dim</em><em>)</em>) – samples in the source domain</p></li>
<li><p><strong>X_t</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_b</em><em>, </em><em>dim</em><em>)</em>) – samples in the target domain</p></li>
<li><p><strong>a</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_a</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – samples weights in the source domain</p></li>
<li><p><strong>b</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_b</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – samples weights in the target domain</p></li>
<li><p><strong>n_projections</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – Number of projections used for the Monte-Carlo approximation</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional =</em>) – Power p used for computing the sliced Wasserstein</p></li>
<li><p><strong>projections</strong> (<em>shape</em><em> (</em><em>dim</em><em>, </em><em>n_projections</em><em>)</em><em>, </em><em>optional</em>) – Projection matrix (n_projections and seed are not used in this case)</p></li>
<li><p><strong>seed</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em> or </em><em>RandomState</em><em> or </em><em>None</em><em>, </em><em>optional</em>) – Seed used for random number generator</p></li>
<li><p><strong>log</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.14)"><em>bool</em></a><em>, </em><em>optional</em>) – if True, sliced_wasserstein_distance returns the projections used and their associated EMD.</p></li>
<li><p><strong>scaler</strong> (<em>None</em><em>, </em><em>object with .transform</em><em>(</em><em>)</em><em>, or </em><em>callable</em><em>, </em><em>optional</em>) – <p>Preprocessing applied to X_s and X_t before computing the distance.
Useful for normalizing inputs when features have very different scales.</p>
<ul>
<li><p><code class="docutils literal notranslate"><span class="pre">None</span></code> : no preprocessing (default)</p></li>
<li><p>Object with <code class="docutils literal notranslate"><span class="pre">.transform()</span></code> method : e.g. an <a class="reference internal" href="ot.utils.html#id46" title="ot.utils.DataScaler"><code class="xref py py-class docutils literal notranslate"><span class="pre">ot.utils.DataScaler</span></code></a>
fitted on a representative sample. This is the recommended way to get
stable, consistent normalization across multiple calls (e.g. when
using SWD as a loss in mini-batch training).</p></li>
<li><p>Callable : any function, lambda, or PyTorch transform applied
directly as <code class="docutils literal notranslate"><span class="pre">scaler(X_s)</span></code> and <code class="docutils literal notranslate"><span class="pre">scaler(X_t)</span></code>.</p></li>
</ul>
<p>See <a class="reference internal" href="ot.utils.html#id46" title="ot.utils.DataScaler"><code class="xref py py-class docutils literal notranslate"><span class="pre">ot.utils.DataScaler</span></code></a> for a backend-aware scaler that supports
joint fitting on multiple distributions.</p>
</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>cost</strong> (<em>float</em>) – Sliced Wasserstein Cost</p></li>
<li><p><strong>log</strong> (<em>dict, optional</em>) – log dictionary return only if log==True in parameters</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="n">n_samples_a</span> <span class="o">=</span> <span class="mi">20</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="p">(</span><span class="n">n_samples_a</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">sliced_wasserstein_distance</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="go">0.0</span>
</pre></div>
</div>
<p class="rubric">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id4" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>35<span class="fn-bracket">]</span></span>
<p>Deshpande, I., Hu, Y. T., Sun, R., Pyrros, A., Siddiqui, N., Koyejo, S., … & Schwing, A. G. (2019). Max-sliced wasserstein distance and its use for gans. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (pp. 10648-10656).</p>
</aside>
</aside>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.sliced.min_sliced_transport_plan">
<span class="sig-prename descclassname"><span class="pre">ot.sliced.</span></span><span class="sig-name descname"><span class="pre">min_sliced_transport_plan</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X_s</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">X_t</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">metric</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'sqeuclidean'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n_projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">batch_size</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dense</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">log</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/sliced/_sliced_plans.html#min_sliced_transport_plan"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.sliced.min_sliced_transport_plan" title="Link to this definition"></a></dt>
<dd><p>Computes the cost and permutation associated to the min-Pivot Sliced
Discrepancy (introduced as min-SWGG in [85] and studied further in [86]). Given
the supports <cite>X_s</cite> and <cite>X_t</cite> of two discrete uniform measures with <cite>ns</cite> and <cite>nt</cite>
atoms in dimension <cite>d</cite>, the min-Pivot Sliced Discrepancy goes through
<cite>n_projections</cite> different projections of the measures on random directions, and
retains the couplings that yields the lowest cost between <cite>X_s</cite> and <cite>X_t</cite>
(compared in <span class="math notranslate nohighlight">\(\mathbb{R}^d\)</span>). When <cite>ns=nt</cite>, it gives</p>
<div class="math notranslate nohighlight">
\[\mathrm{min\text{-}PS}_p^p(X_s, X_t) \approx
\min_{k \in [1, n_{\mathrm{proj}}]} \left(
\frac{1}{n_s} \sum_{i=1}^{n_s} \|X_{s,i} - X_{t,\sigma_k(i)}\|_2^p \right),\]</div>
<p>where <span class="math notranslate nohighlight">\(\sigma_k\)</span> is a permutation such that ordering the projections
on the axis <cite>projections[k, :]</cite> matches <span class="math notranslate nohighlight">\(X_s[i, :]\)</span> to <span class="math notranslate nohighlight">\(X_t[\sigma_k(i), :]\)</span>.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The computation ignores potential ambiguities in the projections: if
two points from a same measure have the same projection on a direction,
then multiple sorting permutations are possible. To avoid combinatorial
explosion, only one permutation is retained: this strays from theory in
pathological cases.</p>
</div>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>Tensorflow and jax only returns dense plans, as they do not support well sparse matrices.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>X_s</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>ns</em><em>, </em><em>d</em><em>)</em>) – The first set of vectors.</p></li>
<li><p><strong>X_t</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>nt</em><em>, </em><em>d</em><em>)</em>) – The second set of vectors.</p></li>
<li><p><strong>a</strong> (<em>ndarray</em><em> of </em><em>float64</em><em>, </em><em>shape</em><em> (</em><em>ns</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – Source histogram (default is uniform weight)</p></li>
<li><p><strong>b</strong> (<em>ndarray</em><em> of </em><em>float64</em><em>, </em><em>shape</em><em> (</em><em>nt</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – Target histogram (default is uniform weight)</p></li>
<li><p><strong>projections</strong> (<em>shape</em><em> (</em><em>dim</em><em>, </em><em>n_projections</em><em>)</em><em>, </em><em>optional</em>) – Projection matrix (n_projections and seed are not used in this case).
Default is None</p></li>
<li><p><strong>metric</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)"><em>str</em></a><em>, </em><em>optional</em><em> (</em><em>default='sqeuclidean'</em><em>)</em>) – Metric to be used. Only works with either of the strings
<cite>‘sqeuclidean’</cite>, <cite>‘minkowski’</cite>, <cite>‘cityblock’</cite>, or <cite>‘euclidean’</cite>.</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em><em> (</em><em>default=1.0</em><em>)</em>) – The p-norm to apply for if metric=’minkowski’</p></li>
<li><p><strong>n_projections</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – The number of projection directions. Required if projections is None.</p></li>
<li><p><strong>seed</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – The seed for the random number generator for sampling projections, in case
projections is None. Default is None.</p></li>
<li><p><strong>batch_size</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – If specified, compute the distance in batches of size <cite>batch_size</cite> to
avoid memory issues for large datasets. Default is None (no batching).</p></li>
<li><p><strong>dense</strong> (<em>boolean</em><em>, </em><em>optional</em><em> (</em><em>default=True</em><em>)</em>) – If True, returns <span class="math notranslate nohighlight">\(\gamma\)</span> as a dense ndarray of shape (ns, nt).
Otherwise returns a sparse representation using scipy’s <cite>coo_matrix</cite>
format.</p></li>
<li><p><strong>log</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.14)"><em>bool</em></a><em>, </em><em>optional</em>) – If True, returns additional logging information. Default is False.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>plan</strong> (<em>ndarray, shape (ns, nt) or coo_matrix if dense is False</em>) – Optimal transportation matrix for the given parameters.</p></li>
<li><p><strong>cost</strong> (<em>float</em>) – The cost associated to the optimal permutation.</p></li>
<li><p><strong>log_dict</strong> (<em>dict, optional</em>) – A dictionary containing intermediate computations for logging purposes.
Returned only if <cite>log</cite> is True.</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id5" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>85<span class="fn-bracket">]</span></span>
<p>Mahey, G., Chapel, L., Gasso, G., Bonet, C., & Courty, N. (2023).
Fast Optimal Transport through Sliced Generalized Wasserstein
Geodesics. Advances in Neural Information Processing Systems, 36,
35350–35385.</p>
</aside>
<aside class="footnote brackets" id="id6" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>86<span class="fn-bracket">]</span></span>
<p>Tanguy, E., Chapel, L., Delon, J. (2025). Sliced Transport
Plans. arXiv preprint 2506.03661.</p>
</aside>
</aside>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">ot</span>
<span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">3.</span><span class="p">,</span><span class="mf">3.</span><span class="p">],</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span><span class="mf">1.</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">2.</span><span class="p">,</span><span class="mf">2.5</span><span class="p">],</span> <span class="p">[</span><span class="mf">3.</span><span class="p">,</span><span class="mf">2.</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">projections</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">plan</span><span class="p">,</span> <span class="n">cost</span> <span class="o">=</span> <span class="n">ot</span><span class="o">.</span><span class="n">min_sliced_transport_plan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">projections</span><span class="o">=</span><span class="n">projections</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">plan</span>
<span class="go">array([[0. , 0.5],</span>
<span class="go"> [0.5, 0. ]])</span>
<span class="gp">>>> </span><span class="n">cost</span>
<span class="go">2.125</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.sliced.projection_sphere_to_circle">
<span class="sig-prename descclassname"><span class="pre">ot.sliced.</span></span><span class="sig-name descname"><span class="pre">projection_sphere_to_circle</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n_projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">50</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">backend</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/sliced/_utils.html#projection_sphere_to_circle"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.sliced.projection_sphere_to_circle" title="Link to this definition"></a></dt>
<dd><p>Projection of <span class="math notranslate nohighlight">\(x\in S^{d-1}\)</span> on circles using coordinates on [0,1[.</p>
<p>To get the projection on the circle, we use the following formula:</p>
<div class="math notranslate nohighlight">
\[P^U(x) = \frac{U^Tx}{\|U^Tx\|_2}\]</div>
<p>where <span class="math notranslate nohighlight">\(U\)</span> is a random matrix sampled from the uniform distribution on the Stiefel manifold of dimension <span class="math notranslate nohighlight">\(d\times 2\)</span>: <span class="math notranslate nohighlight">\(\mathbb{V}_{d,2}=\{X \in \mathbb{R}^{d\times 2}, X^TX=I_2\}\)</span>
and <span class="math notranslate nohighlight">\(x\)</span> is a point on the sphere. Then, we apply the function get_coordinate_circle to get the coordinates on <span class="math notranslate nohighlight">\([0,1[\)</span>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>x</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples</em><em>, </em><em>dim</em><em>)</em>) – samples on the sphere</p></li>
<li><p><strong>n_projections</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – Number of projections used for the Monte-Carlo approximation</p></li>
<li><p><strong>projections</strong> (<em>shape</em><em> (</em><em>n_projections</em><em>, </em><em>dim</em><em>, </em><em>2</em><em>)</em><em>, </em><em>optional</em>) – Projection matrix (n_projections and seed are not used in this case)</p></li>
<li><p><strong>seed</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em> or </em><em>RandomState</em><em> or </em><em>None</em><em>, </em><em>optional</em>) – Seed used for random number generator</p></li>
<li><p><strong>backend</strong> – Backend to use for random generation</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><strong>Xp_coords</strong> – Coordinates of the projections on the circle</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>ndarray, shape (n_projections, n_samples)</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.sliced.sliced_plans">
<span class="sig-prename descclassname"><span class="pre">ot.sliced.</span></span><span class="sig-name descname"><span class="pre">sliced_plans</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X_s</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">X_t</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">metric</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'sqeuclidean'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n_projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">batch_size</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">log</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/sliced/_sliced_plans.html#sliced_plans"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.sliced.sliced_plans" title="Link to this definition"></a></dt>
<dd><p>Computes all the permutations that sort the projections of two <cite>(ns, nt)</cite>
datasets <cite>X_s</cite> and <cite>X_t</cite> on the directions <cite>projections</cite>.
Each permutation <cite>perm[:, k]</cite> is such that each <span class="math notranslate nohighlight">\(X_s[i, :]\)</span> is matched
to <cite>X_t[perm[i, k], :]</cite> when projected on <cite>projections[k, :]</cite>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>X_s</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>ns</em><em>, </em><em>d</em><em>)</em>) – The first set of vectors.</p></li>
<li><p><strong>X_t</strong> (<em>array-like</em><em>, </em><em>shape</em><em> (</em><em>nt</em><em>, </em><em>d</em><em>)</em>) – The second set of vectors.</p></li>
<li><p><strong>a</strong> (<em>ndarray</em><em> of </em><em>float64</em><em>, </em><em>shape</em><em> (</em><em>ns</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – Source histogram (default is uniform weight)</p></li>
<li><p><strong>b</strong> (<em>ndarray</em><em> of </em><em>float64</em><em>, </em><em>shape</em><em> (</em><em>nt</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – Target histogram (default is uniform weight)</p></li>
<li><p><strong>metric</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.14)"><em>str</em></a><em>, </em><em>optional</em><em> (</em><em>default='sqeuclidean'</em><em>)</em>) – Metric to be used. Only works with either of the strings
<cite>‘sqeuclidean’</cite>, <cite>‘minkowski’</cite>, <cite>‘cityblock’</cite>, or <cite>‘euclidean’</cite>.</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em><em> (</em><em>default=1.0</em><em>)</em>) – The p-norm to apply for if metric=’minkowski’</p></li>
<li><p><strong>projections</strong> (<em>shape</em><em> (</em><em>dim</em><em>, </em><em>n_projections</em><em>)</em><em>, </em><em>optional</em>) – Projection matrix (n_projections and seed are not used in this case)</p></li>
<li><p><strong>n_projections</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – The number of projection directions. Required if projections is None.</p></li>
<li><p><strong>seed</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – The seed for the random number generator for sampling projections, in case
projections is None.
Default is None.</p></li>
<li><p><strong>batch_size</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – If specified, compute the distance in batches of size <cite>batch_size</cite> to
avoid memory issues for large datasets. Default is None (no batching).</p></li>
<li><p><strong>log</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.14)"><em>bool</em></a><em>, </em><em>optional</em>) – If True, returns additional logging information. Default is False.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>plan</strong> (<em>list of dictionaries</em>) – List of the optimal transport plans as a list of dictionaries containing
the rows, cols and data of the non-zero elements of the transportation matrix.</p></li>
<li><p><strong>costs</strong> (<em>list of float</em>) – The cost associated to each projection.</p></li>
<li><p><strong>log_dict</strong> (<em>dict, optional</em>) – A dictionary containing intermediate computations for logging purposes.
Returned only if <cite>log</cite> is True.</p></li>
</ul>
</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.sliced.sliced_wasserstein_distance">
<span class="sig-prename descclassname"><span class="pre">ot.sliced.</span></span><span class="sig-name descname"><span class="pre">sliced_wasserstein_distance</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X_s</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">X_t</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n_projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">50</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">log</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">scaler</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/sliced/_sliced_distances.html#sliced_wasserstein_distance"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.sliced.sliced_wasserstein_distance" title="Link to this definition"></a></dt>
<dd><p>Computes a Monte-Carlo approximation of the p-Sliced Wasserstein distance</p>
<div class="math notranslate nohighlight">
\[\mathcal{SWD}_p(\mu, \nu) = \underset{\theta \sim \mathcal{U}(\mathbb{S}^{d-1})}{\mathbb{E}}\left(\mathcal{W}_p^p(\theta_\# \mu, \theta_\# \nu)\right)^{\frac{1}{p}}\]</div>
<p>where :</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(\theta_\# \mu\)</span> stands for the pushforwards of the projection <span class="math notranslate nohighlight">\(X \in \mathbb{R}^d \mapsto \langle \theta, X \rangle\)</span></p></li>
</ul>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>X_s</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_a</em><em>, </em><em>dim</em><em>)</em>) – samples in the source domain</p></li>
<li><p><strong>X_t</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_b</em><em>, </em><em>dim</em><em>)</em>) – samples in the target domain</p></li>
<li><p><strong>a</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_a</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – samples weights in the source domain</p></li>
<li><p><strong>b</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_b</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – samples weights in the target domain</p></li>
<li><p><strong>n_projections</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – Number of projections used for the Monte-Carlo approximation</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em>) – Power p used for computing the sliced Wasserstein</p></li>
<li><p><strong>projections</strong> (<em>shape</em><em> (</em><em>dim</em><em>, </em><em>n_projections</em><em>)</em><em>, </em><em>optional</em>) – Projection matrix (n_projections and seed are not used in this case)</p></li>
<li><p><strong>seed</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em> or </em><em>RandomState</em><em> or </em><em>None</em><em>, </em><em>optional</em>) – Seed used for random number generator</p></li>
<li><p><strong>log</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.14)"><em>bool</em></a><em>, </em><em>optional</em>) – if True, sliced_wasserstein_distance returns the projections used and their associated EMD.</p></li>
<li><p><strong>scaler</strong> (<em>None</em><em>, </em><em>object with .transform</em><em>(</em><em>)</em><em>, or </em><em>callable</em><em>, </em><em>optional</em>) – <p>Preprocessing applied to X_s and X_t before computing the distance.
Useful for normalizing inputs when features have very different scales.</p>
<ul>
<li><p><code class="docutils literal notranslate"><span class="pre">None</span></code> : no preprocessing (default)</p></li>
<li><p>Object with <code class="docutils literal notranslate"><span class="pre">.transform()</span></code> method : e.g. an <a class="reference internal" href="ot.utils.html#id46" title="ot.utils.DataScaler"><code class="xref py py-class docutils literal notranslate"><span class="pre">ot.utils.DataScaler</span></code></a>
fitted on a representative sample. This is the recommended way to get
stable, consistent normalization across multiple calls (e.g. when
using SWD as a loss in mini-batch training).</p></li>
<li><p>Callable : any function, lambda, or PyTorch transform applied
directly as <code class="docutils literal notranslate"><span class="pre">scaler(X_s)</span></code> and <code class="docutils literal notranslate"><span class="pre">scaler(X_t)</span></code>.</p></li>
</ul>
<p>See <a class="reference internal" href="ot.utils.html#id46" title="ot.utils.DataScaler"><code class="xref py py-class docutils literal notranslate"><span class="pre">ot.utils.DataScaler</span></code></a> for a backend-aware scaler that supports
joint fitting on multiple distributions.</p>
</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>cost</strong> (<em>float</em>) – Sliced Wasserstein Cost</p></li>
<li><p><strong>log</strong> (<em>dict, optional</em>) – log dictionary return only if log==True in parameters</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="n">n_samples_a</span> <span class="o">=</span> <span class="mi">20</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="p">(</span><span class="n">n_samples_a</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">sliced_wasserstein_distance</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="go">0.0</span>
</pre></div>
</div>
<p class="rubric">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id7" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>31<span class="fn-bracket">]</span></span>
<p>Bonneel, Nicolas, et al. “Sliced and radon wasserstein barycenters of measures.” Journal of Mathematical Imaging and Vision 51.1 (2015): 22-45</p>
</aside>
</aside>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.sliced.sliced_wasserstein_sphere">
<span class="sig-prename descclassname"><span class="pre">ot.sliced.</span></span><span class="sig-name descname"><span class="pre">sliced_wasserstein_sphere</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X_s</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">X_t</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n_projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">50</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">log</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/sliced/_spherical_sliced.html#sliced_wasserstein_sphere"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.sliced.sliced_wasserstein_sphere" title="Link to this definition"></a></dt>
<dd><p>Compute the spherical sliced-Wasserstein discrepancy.</p>
<div class="math notranslate nohighlight">
\[SSW_p(\mu,\nu) = \left(\int_{\mathbb{V}_{d,2}} W_p^p(P^U_\#\mu, P^U_\#\nu)\ \mathrm{d}\sigma(U)\right)^{\frac{1}{p}}\]</div>
<p>where:</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(P^U_\# \mu\)</span> stands for the pushforwards of the projection <span class="math notranslate nohighlight">\(\forall x\in S^{d-1},\ P^U(x) = \frac{U^Tx}{\|U^Tx\|_2}\)</span></p></li>
</ul>
<p>The function runs on backend but tensorflow and jax are not supported.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>X_s</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_a</em><em>, </em><em>dim</em><em>)</em>) – Samples in the source domain</p></li>
<li><p><strong>X_t</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_b</em><em>, </em><em>dim</em><em>)</em>) – Samples in the target domain</p></li>
<li><p><strong>a</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_a</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – samples weights in the source domain</p></li>
<li><p><strong>b</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_b</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – samples weights in the target domain</p></li>
<li><p><strong>n_projections</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – Number of projections used for the Monte-Carlo approximation</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.14)"><em>float</em></a><em>, </em><em>optional</em><em> (</em><em>default=2</em><em>)</em>) – Power p used for computing the spherical sliced Wasserstein</p></li>
<li><p><strong>projections</strong> (<em>shape</em><em> (</em><em>n_projections</em><em>, </em><em>dim</em><em>, </em><em>2</em><em>)</em><em>, </em><em>optional</em>) – Projection matrix (n_projections and seed are not used in this case)</p></li>
<li><p><strong>seed</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em> or </em><em>RandomState</em><em> or </em><em>None</em><em>, </em><em>optional</em>) – Seed used for random number generator</p></li>
<li><p><strong>log</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.14)"><em>bool</em></a><em>, </em><em>optional</em>) – if True, sliced_wasserstein_sphere returns the projections used and their associated EMD.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>cost</strong> (<em>float</em>) – Spherical Sliced Wasserstein Cost</p></li>
<li><p><strong>log</strong> (<em>dict, optional</em>) – log dictionary return only if log==True in parameters</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">ot</span>
<span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="n">n_samples_a</span> <span class="o">=</span> <span class="mi">20</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="p">(</span><span class="n">n_samples_a</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="n">X</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">X</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">ot</span><span class="o">.</span><span class="n">sliced_wasserstein_sphere</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="go">0.0</span>
</pre></div>
</div>
<p class="rubric">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id8" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>46<span class="fn-bracket">]</span></span>
<p>Bonet, C., Berg, P., Courty, N., Septier, F., Drumetz, L., & Pham, M. T. (2023). Spherical sliced-wasserstein. International Conference on Learning Representations.</p>
</aside>
</aside>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="ot.sliced.sliced_wasserstein_sphere_unif">
<span class="sig-prename descclassname"><span class="pre">ot.sliced.</span></span><span class="sig-name descname"><span class="pre">sliced_wasserstein_sphere_unif</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X_s</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n_projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">50</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">projections</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">seed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">log</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../_modules/ot/sliced/_spherical_sliced.html#sliced_wasserstein_sphere_unif"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#ot.sliced.sliced_wasserstein_sphere_unif" title="Link to this definition"></a></dt>
<dd><p>Compute the 2-spherical sliced wasserstein w.r.t. a uniform distribution.</p>
<div class="math notranslate nohighlight">
\[SSW_2(\mu_n, \nu)\]</div>
<p>where</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(\mu_n=\sum_{i=1}^n \alpha_i \delta_{x_i}\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(\nu=\mathrm{Unif}(S^{d-1})\)</span></p></li>
</ul>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>X_s</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_a</em><em>, </em><em>dim</em><em>)</em>) – Samples in the source domain</p></li>
<li><p><strong>a</strong> (<em>ndarray</em><em>, </em><em>shape</em><em> (</em><em>n_samples_a</em><em>,</em><em>)</em><em>, </em><em>optional</em>) – samples weights in the source domain</p></li>
<li><p><strong>n_projections</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em>, </em><em>optional</em>) – Number of projections used for the Monte-Carlo approximation</p></li>
<li><p><strong>projections</strong> (<em>shape</em><em> (</em><em>n_projections</em><em>, </em><em>dim</em><em>, </em><em>2</em><em>)</em><em>, </em><em>optional</em>) – Projection matrix (n_projections and seed are not used in this case)</p></li>
<li><p><strong>seed</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.14)"><em>int</em></a><em> or </em><em>RandomState</em><em> or </em><em>None</em><em>, </em><em>optional</em>) – Seed used for random number generator</p></li>
<li><p><strong>log</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.14)"><em>bool</em></a><em>, </em><em>optional</em>) – if True, sliced_wasserstein_distance returns the projections used and their associated EMD.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>cost</strong> (<em>float</em>) – Spherical Sliced Wasserstein Cost</p></li>
<li><p><strong>log</strong> (<em>dict, optional</em>) – log dictionary return only if log==True in parameters</p></li>
</ul>
</p>
</dd>
</dl>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">ot</span>
<span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">x0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">500</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">x0</span> <span class="o">=</span> <span class="n">x0</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">x0</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">ssw</span> <span class="o">=</span> <span class="n">ot</span><span class="o">.</span><span class="n">sliced_wasserstein_sphere_unif</span><span class="p">(</span><span class="n">x0</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">ot</span><span class="o">.</span><span class="n">sliced_wasserstein_sphere_unif</span><span class="p">(</span><span class="n">x0</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">42</span><span class="p">),</span> <span class="mf">0.01734</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
<section id="references">
<h2>References:<a class="headerlink" href="#references" title="Link to this heading"></a></h2>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id9" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>46<span class="fn-bracket">]</span></span>
<p>Bonet, C., Berg, P., Courty, N., Septier, F., Drumetz, L., & Pham, M. T. (2023). Spherical sliced-wasserstein. International Conference on Learning Representations.</p>
</aside>
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